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Gather all ingredients.
Preheat the oven to 300 degrees F (150 degrees C).
Heat olive oil in an oven-proof deep skillet with a lid over medium-high heat. Season beef shank with salt and pepper.
Cook beef shank in hot oil until browned on all sides, 5 to 10 minutes. Transfer beef shank to a plate.
Sauté onion and celery in the same skillet until onion turns translucent, 5 to 10 minutes.
Pour in wine and bring to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon.
Place beef shank into onion-wine mixture; add beef broth and pour in enough water to cover ½ to ¾ of the beef shank. Sprinkle tarragon over shank; cover the skillet with a lid.
Bake in the preheated oven until meat falls off the bone, about 5 hours. Season with salt and pepper before serving.